Initial value problem of Oshida-Rule

Let $m$ be a positive integer, let $e$ denote the distinguished initial vector, and let $h$, $h'$, $f_1$, and $H_{245}=L\circ S$ be the maps defined above, with $u\in\mathbb{R}^{n}_{\operatorname{max}}$. Initial value problem of Oshida-Rule. For every non-negative integer $n$ and every vector $u\in\mathbb{R}^{n}_{\operatorname{max}}$ with $u\neq e$,

$$h' \circ h \circ \underbrace{f_1\circ\cdots\circ f_1}_{2m-1}(u)\neq \underbrace{H_{245}\circ\cdots\circ H_{245}}_{2m-1}\circ h'\circ h(u).$$

Equivalently,

$$\left\{u\in\mathbb{R}^{n}_{\operatorname{max}}\mathrel{\middle|}h'\circ h\circ\underbrace{f_1\circ\cdots\circ f_1}_{2m-1}(u)=\underbrace{H_{245}\circ\cdots\circ H_{245}}_{2m-1}\circ h'\circ h(u)\right\}=\{e\}.$$

This is the proposed uniqueness statement for the initial value of the Oshida cellular-automaton rule. The supplied text gives no evidence that the conjecture has been proved or disproved.